The Chebyshev(ODE, y(x)) command finds the solution of a Chebyshev equation, which is a linear homogeneous ordinary differential equation of the form:

Use the option output=steps to make this command return an annotated step-by-step solution.  Further control over the format and display of the step-by-step solution is available using the options described in Student:-Basics:-OutputStepsRecord.  The options supported by that command can be passed to this one.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\left[\mathrm{Solve}\right]\right)\:$

$\mathrm{ode1}≔\left(-x^2+1\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-x\mathrm{diff}\left(y\left(x\right)\,x\right)+y\left(x\right)=0$

$\mathrm{ode1}≔\left(-x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+y\left(x\right)=0$

$\mathrm{Chebyshev}\left(\mathrm{ode1}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{\_C1}\sqrt{-x^2+1}+\mathrm{\_C2}x$

$\mathrm{ode2}≔\left(-x^2+1\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-x\mathrm{diff}\left(y\left(x\right)\,x\right)+4y\left(x\right)=0$

$\mathrm{ode2}≔\left(-x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+4y\left(x\right)=0$

$\mathrm{Chebyshev}\left(\mathrm{ode2}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{\_C1}x\sqrt{-x^2+1}+\mathrm{\_C2}\left(2x^2-1\right)$

$\mathrm{ode3}≔\left(-x^2+1\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-x\mathrm{diff}\left(y\left(x\right)\,x\right)+9y\left(x\right)=0$

$\mathrm{ode3}≔\left(-x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+9y\left(x\right)=0$

$\mathrm{Chebyshev}\left(\mathrm{ode3}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{\_C1}\sin \left(3\arccos \left(x\right)\right)+\mathrm{\_C2}\cos \left(3\arccos \left(x\right)\right)$

$\mathrm{ode4}≔\left(-x^2+1\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-x\mathrm{diff}\left(y\left(x\right)\,x\right)-25y\left(x\right)=0$

$\mathrm{ode4}≔\left(-x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-25y\left(x\right)=0$

$\mathrm{Chebyshev}\left(\mathrm{ode4}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{\_C1}{\ⅇ}^{5\arccos \left(x\right)}+\mathrm{\_C2}{\ⅇ}^{-5\arccos \left(x\right)}$

The Student[ODEs][Solve][Chebyshev] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.

The Student[ODEs][Solve][Chebyshev] command was updated in Maple 2022.

The output option was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.